2D HILBERT-HUANG TRANSFORM

Jeremy Schmitt, Nelly Pustelnik, Pierre Borgnat, Patrick Flandrin

Laboratoire de Physique de l’Ecole Normale Suprieure de Lyon, CNRS and Universite de Lyon, Francefirst.last@ens-lyon.fr

ABSTRACTThis paper presents a 2D transposition of the Hilbert-HuangTransform (HHT), an empirical data analysis method de-signed for studying instantaneous amplitudes and phases ofnon-stationary data. The principle is to adaptively decom-pose an image into oscillating parts called Intrinsic ModeFunctions (IMFs) using an Empirical Mode Decompositionmethod (EMD), and them to perform Hilbert spectral anal-ysis on the IMFs in order to recover local amplitudes andphases. For the decomposition step, we propose a new 2Dmode decomposition method based on non-smooth convexoptimization, while for the instantaneous spectral analysis,we use a 2D transposition of Hilbert spectral analysis calledmonogenic analysis, based on Riesz transform and allowingto extract instantaneous amplitudes, phases, and orientations.The resulting 2D-HHT is validated on simulated data.

Index Terms— Hilbert-Huang Transform, empirical mode de-composition, convex optimization, proximal algorithms, Riesz trans-form, monogenic analysis

1. INTRODUCTION

The 1D Hilbert-Huang Transform (1D-HHT), introduced byHuang et al. [1], is an empirical method for data analysis.Compared to usual time-frequency/time-scale method such aswavelet analysis or Wigner-Ville distribution, which aims atanalysing non-linear and non-stationary signals, this methodfavours adaptivity.

Formally, the objective of 1D-HHT is to extract the instan-taneous amplitudes (α(k))1≤k≤K and phases (ξ(k))1≤k≤Kfrom a signal x ∈ RN built as a sum of elementary functions(d(k))1≤k≤K oscillating around zero, called Intrinsic ModeFunctions (IMFs), and a trend a(k) ∈ RN , i.e.,

x = a(k) +

K∑k=1

α(k) cos ξ(k)︸ ︷︷ ︸d(k)

.

To achieve this goal, the 1D-HHT consists in a two-stepprocedure combining (i) a decomposition step, whose objec-tive is to extract the IMFs (d(k))1≤k≤K from the data x, with(ii) a Hilbert spectral analysis of each extracted IMF d(k)

This work was supported by the Agence Nationale de la Recherche undergrant ANR-13-BS03-0002-01

in order to estimate the instantaneous amplitudes α(k) andphases ξ(k). Regarding the first step, an efficient decompo-sition procedure known as Empirical Mode Decomposition(EMD) has been proposed in [1]. It aims at sequentially ex-tracting the IMF d(k) from a temporary trend a(k−1), definedas {

a(0) = x,

(∀k ∈ {1, . . . ,K}) a(k−1) = a(k) + d(k),

through a sifting process that is based on the computation ofa mean envelope of a(k−1) (mean of the upper and lower en-velopes obtained by interpolating the maxima, resp. minima,of a(k−1)).

The aim of this work is to propose the counterpart ofthe Hilbert-Huang transform for image analysis in order todecompose an image into elementary components and extracttheir instantaneous amplitudes, phases, and orientations. Af-ter a short review of related works in Section 2, we detail theproposed 2D-HHT method in Section 3. In Section 4, weillustrate the efficiency of the proposed method compared tothe state-of-the art techniques on simulated data.

Notations We denote y = (yn,m)1≤n≤N1,1≤m≤N2∈ RN1×N2

the matrix expression of an image whose size is N1 × N2,the n-th row of the image y is denoted yn,• ∈ RN2 , andy = (yn)1≤n≤N ∈ RN is the vector expression of y, suchthat N = N1 ×N2.

2. RELATED WORKS

The Riesz-Laplace transform proposed in [2], which consistsin a multiresolution 2D spectral analysis method, refers to themethod with the closest goal to 2D-HHT. More precisely, thismethod aims at combining a two-dimensional wavelet trans-form with a monogenic analysis [3], which is based on Rieszanalysis and appears to be a 2D extension of the Hilbert trans-form. The counterpart of using a wavelet framework is thelack of adaptivity and consequently this method is less suitedfor analysing non-stationary signals such as AM-FM signals.To build a genuine 2D-HHT method, a solution is to combineFelsberg and Sommer monogenic analysis [3] with a robust2D-EMD. However, to the best of our knowledge, there is nosuch a solution in the literature.

Before detailing the proposed 2D-HHT, we will discussthe robustness of the existing EMD methods in order tohighlight the necessity to propose a new robust 2D mode de-composition procedure. On the one hand, existing 2D-EMDmethods are based on the sifting procedure whose main draw-back is the lack of a rigorous mathematical definition, andconsequently of convergence properties [4, 5, 6, 7, 8, 9]. Onthe other hand, efficient 1D mode decomposition proceduresbased on convex optimization have been recently proposedin order to get stronger mathematical guarantees [10, 11, 12].For instance, [12] proposed a mathematical formalism forEMD based on a multicomponent proximal algorithm thatcombines the principle of texture-geometry decomposition[13, 14, 15] with some specific features of EMD: constraintson extrema in order to extract IMFs oscillating around zero,sequential aspect of EMD, or IMFs quasi-orthogonality. Thismethods appears to have better performance (in terms ofextraction or convergence guarantees) than the other convexoptimization procedures as discussed in [10, 11]. For this rea-son, we propose to extend this method to a 2D mode decom-position formalism and thus to combine it with a monogenicanalysis in order to build a complete 2D-HHT.

3. PROPOSED 2D-HHT

3.1. 2D proximal mode decomposition

As discussed previously, the mode decomposition aims atsplitting up the trend a(k−1) into a component having IMFproperties (i.e. d(k)), and a residual component, denoted a(k).To obtain such a decomposition we propose to solve

(a(k), d(k)) ∈ Argmina∈RN ,d∈RN

φk(a) + ψk(d) + ϕk(a, d; a(k−1))

where φk and ψk denote convex, lower semi-continuous, andproper functions from RN to ]−∞,+∞] that impose the trendand IMF behaviour to the components a(k) and d(k) respec-tively, while ϕk(·, ·; a(k−1)) denotes a convex, lower semi-continuous, proper function from RN × RN to ] −∞,+∞]that aims to model that a(k−1) is close to a(k) + d(k).

The smoothness of the k-order trend is obtained by im-posing a constraint on its total variation, i.e.,

φk(a)=η(k)

N1∑n=1

N2∑m=1

√|an−1,m − an,m|2+|an,m−1 − an,m|2

with η(k) > 0.The tricky step in order to propose a 2D extension of

the 1D proximal decomposition procedure [12] lies in thedefinition of the zero mean envelope constraint through thefunction ψ. Here we propose a ”Pseudo 2D-EMD” approach,where lines, columns and diagonals extrema are separatelyconstrained. For instance, the extrema-based constraint canbe written for each row n ∈ {1, . . . , N1}, ‖R(k)

n dn,•‖1,

where R(k)n ∈ RN2×N2 denotes the linear combination of

some elements of dn,• allowing to impose a zero mean enve-lope of the component d(k) (cf. [12] for the construction ofR

(k)n that is similar due to the fact that a row dn,• behaves like

a 1D signal). Considering the entire image, the constraint canbe written ‖R(k)d‖1 where R(k) = diag(R(k)

1 , . . . , R(k)N1

) is ablock diagonal matrix in RN×N , which is highly sparse. Weapply the same type of constraint to the columns (C(k)), thediagonals (D(k)), and the anti-diagonals (A(k)) of the image,leading to the penalization:

ψk(d) =

4∑i=1

λ(k)i ‖M

(k)i d‖1

where M (k)1 = R(k), M (k)

2 = C(k), M (k)3 = D(k), M (k)

4 =A(k) denote matrices in RN×N .

As proposed in [16], the coupling term is chosen quadratic:

ϕk(a, d; a(k−1)) = ‖a+ d− a(k−1)‖2.

The solution of the resulting minimization problem is ob-tained with Condat-Vu primal-dual splitting algorithm [17]that allows to deal with linear operators and non-smooth func-tionals. The iterations are specified in Algorithm 1. For fur-ther details on the algorithmic solution and proximal tools,one may refer to [16].

3.2. Monogenic analysis of the extracted IMFs

Given a real-valued 1D signal y, the associated analytic sig-nal ya(t), which by definition involves the signal itself andits Hilbert transform, can also be written under a polar forminvolving instantaneous phase and amplitude respectively de-noted ξ and α such as: ya = y + jH(y) = αejξ. These twoformulations allow to easily compute the instantaneous am-plitude and the instantaneous phase as the absolute value ofthe analytic signal and its argument.

The Riesz transform is the natural 2D extension of theHilbert transform [3]. The Riesz transform of a 2D signal ycan be expressed as yR = (y(1),y(2)) = (h(1) ∗ y, h(2) ∗ y),where the filters (h(i))1≤i≤2 are characterized by their 2Dtransfer functions H(i)

ω = −jωi/‖ω‖ with ω = (ω1, ω2).On the other hand, the counterpart of the analytic signal in2D is called the monogenic signal. It consists in the three-component signal defined by ym = (y,y(1),y(2)) [3]. In asimilar way to the analytic signal, the monogenic signal en-ables to compute easily the local amplitude, phase, and orien-tation at each pixel through the relations, for every (n,m) ∈{1, . . . , N1} × {1, . . . , N2},

αn,m =

√(yn,m

)2+(y(1)n,m

)2+(y(2)n,m

)2(1)

ξn,m = arctan

(√(y(1)n,m

)2+(y(2)n,m

)2yn,m

)(2)

θn,m = arctan(y(2)n,m/y

(1)n,m). (3)

However, the estimation of the orientation proposed in (3)lacks of robustness because it does not take into account theorientation of neighbouring pixels. Unser et al. [2] derivedan improved estimation based on a minimization procedureincluding a smoothness neighbourhood constraint. In our ex-periments, this robust technique is used.

3.3. 2D-HHT Algorithm

We now summarize the proposed 2D-HHT. In order tolighten the notations, we rewrite the total variation penal-ization as φk = η(k)‖L · ‖2,1, with L = [H∗V ∗]∗ whereH and V denote the operators associated to the horizon-tal and vertical finite differences. We denote M (k) =diag(M (k)

1 ,M(k)2 ,M

(k)3 ,M

(k)4 ). Parameters σ and τ are

chosen so as to ensure the convergence of the algorithm,see [17] for details. The 2D-HHT method is summed up inAlgorithm 1.

Algorithm 1 2D-HHT AlgorithmSTEP 1 – Initialization Set a(0) = x,

Choose the number of IMFs K to be extracted,Set k = 1.

STEP 2 – 2D prox. mode decomp. : extract a(k) and d(k) from a(k−1).

Compute (M(k)i )1≤i≤4 from a(k−1),

Compute β = 1 + ‖M(k)‖2,Set σ > 0 and let τ = 0.9/(σβ + 2),Initialize a[0] and d[0] in RN ,Initialize y[`]0 in R2N and y[`]i ∈ RN for i = 1, · · · , 4.For ` = 0, 1, · · ·∣∣ a[`+1] = a[`] − 2τ(a[`] + d[`] − a(k−1))− τL∗y[`]0∣∣ d[`+1] = d[`] − 2τ(a[`] + d[`] − a(k−1))− τ

∑4i=1M

∗k,iy

[`]i∣∣ y[`+1]

0 = proxσ(η(k)‖·‖1,2)∗ (y[`]0 + σL(2a[`+1] − a[`]))∣∣ For i = 1, · · · , 4⌊

by[`+1]i = prox

σ(λ(k)i ‖·‖1)

∗ (y[`]i + σMk,i(2a

[`+1] − a[`]))

Set d(k) = lim`→∞ d[`] and a(k) = lim`→∞ a[`].STEP 3 – Monogenic Analysis

Compute monogenic signal of d(k) : d(k)m = (d(k),d(k,1),d(k,2))

Compute local amplitude α(k), phase ξ(k) and orientation θ(k)

using Eqs (1), (2), (3). (θ(k) can also be computed using Unser’simproved method).

While k < K, set k ← k + 1 and return to STEP 2.

4. EXPERIMENTS

The image to be analyzed (Figure 1) consists in a sum of atrend and two localized texture components x(1) and x(2).The trend is formed with one rectangular patch and one el-lipsoidal patch. The component x(1) (resp. x(2)) models amodulated signal of central frequency f1 = 120/512 (resp.f2 = 60/512).

We apply the proposed 2D-HHT method to extract thetwo resulting IMFs and their local orientation estimates. Inorder to fairly compare with the state-of-the-art methods, wepropose to replace the STEP 2 in Algorithm 1 with other de-composition methods such that Image Empirical Mode De-

composition [5], a natural 2D extension of 1D EMD based on2D interpolation of extrema using thin-plate splines, and twotexture-cartoon decomposition methods that are total varia-tion and Gilles-Osher texture-geometry decomposition [14].On the other hand we compare the extracted orientation withthe results obtained from the Riesz-Laplace analysis proposedin [2]. Total variation decomposition consists in solving theoptimization problem Argmina∈RN ‖a − a(k−1)‖22 + φk(a),where φk is the total variation constraint as defined in 3.1.Gilles-Osher is an iterative algorithm designed for solvingMeyer’s G-norm texture-cartoon decomposition model (wedenote µ(k) the texture regularization parameter and λ(k) thecartoon regularization parameter). We use the following opti-mal regularization parameters for our 2D-HHT : η(1) = 0.3,λ(1)i ≡ 0.3, η(2) = 1, λ(2)i ≡ 0.1. For Total Variation de-

composition method, we use : η(1) = 70 and η(1) = 100.For Gilles-Osher method, we set µ(1) = 104, λ(1) = 103,µ(2) = 10, and λ(2) = 10. Results are shown on Figs 2 and 3.

First of all, our method provides a good separation of thedifferent components. It has the expected behaviour of a 2D-HHT: the locally fastest oscillating components are extractedat each step of the decomposition, even if their frequenciesare non stationary. Our proposed 2D-EMD method proved toperform better than previous 2D-EMD methods. For instance,the IEMD does not manage to separate at all the componentsx1 and x2. In comparison with other approaches like waveletdecomposition and texture-cartoon decomposition, the 2D-EMD approach provides more adaptivity and a better man-agement of non-stationary signals. The Total Variation basedapproach does not give a good separation of the three oscillat-ing components. Gilles-Osher solution is not suited for non-stationary signals: some of the slower part of the frequencymodulated component x2 is on the 2nd IMF, while its fasterpart are localized on the first IMF. Finally, the Riesz-Laplacesolution provides good results but this method is not adaptiveand less suited for non-stationary signals. For instance, wecan observe that some area of the same component but withdifferent frequencies end up on different wavelet scales : here,the faster part of x2 is on the first scale, while the rest is onthe second and third scale.

5. CONCLUSION

This paper proposes a complete 2D-HTT to extract local am-plitudes, phases, and orientations of non-stationary images.This method is based on a 2D variational mode decomposi-tion combined with a monogenic analysis using Riesz trans-form. This method has been tested on simulated data. In termof local orientation analysis, the proposed proved to be moreefficient than existing 2D EMD methods and more adaptivethan other decomposition approaches. In a future work, wewill focus on the study of instantaneous frequencies consid-ering the derivation of the local phase along the direction pro-vided by the orientation.

Data x(1) x(1) orientations x(2) x(2) orientations orientations colorbar

−1.5

−1

−0.5

0

0.5

1

1.5

Fig. 1. Simulated data and its components : texture components x(1), x(2) and their local orientations.

1st IMF: d(1) 1st IMF orientation: θ(1) 2nd IMF: d(2) 2nd IMF orientation: θ(2) Residual a(2)

Fig. 2. Decomposition and local orientation of the simulated data presented in Fig. 1 obtained with different methods. 1st row:proposed solution, 2nd row: Image Empirical Mode Decomposition, 3rd row: Total Variation based decomposition, and 4throw: Gilles-Osher based decomposition. From the left to the right the columns present d(1), θ(1), d(2), θ(2), and a(2).

1st scale orientation 2nd scale orientation 3rd scale orientation

Fig. 3. Orientations estimated on 3 scales with Riesz-Laplace wavelet transform.

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